8.12 Arima Theory Overview

Recall the components of Arima.

The ARMA Equation

Let $x(t)$ be the time series data, $\alpha$ are the parameters of the AR model

$x(t)-{\alpha }_{1}x(t-1)-\cdots -{\alpha }_{p}x(t-p)=\epsilon (t)+{\theta }_1\epsilon (t-1)+\cdots +{\theta }_q\epsilon (t-q).$

Let $L$ be the lag operator, the above equation is rewritten as

$\left(1-{\sum }_{i=1}^p{\alpha }_i{L}^i\right)x(t)=\left(1-{\sum }_{i=1}^q{\theta }_i{L}^i\right)\epsilon (t).$

The ARIMA Equation

When differencing, we get $\left(1-{\sum }_{i=1}^p{\alpha }_i{L}^i\right)(1-L{)}^{d}x(t)=\left(1-{\sum }_{i=1}^q{\theta }_i{L}^i\right)\epsilon (t).$